- #1
Oxymoron
- 870
- 0
If I have a faithful nondegenerate representation of a C*-algebra, A:
[tex]\pi\,:\,A \rightarrow B(\mathcal{H})[/tex]
where [itex]B(\mathcal{H})[/itex] is the set of all bounded linear operators on a Hilbert space. And just suppose that [itex]a\geq 0 \in A[/itex]. How is the fact that a is positive got anything to do with [itex]\pi(a)[/itex] being positive?
Apparantly there is an if and only if relationship!? How does one begin to prove something like [itex]a\geq 0 \Leftrightarrow \pi(a) \geq 0 \in B(\mathcal{H})[/itex]?
[tex]\pi\,:\,A \rightarrow B(\mathcal{H})[/tex]
where [itex]B(\mathcal{H})[/itex] is the set of all bounded linear operators on a Hilbert space. And just suppose that [itex]a\geq 0 \in A[/itex]. How is the fact that a is positive got anything to do with [itex]\pi(a)[/itex] being positive?
Apparantly there is an if and only if relationship!? How does one begin to prove something like [itex]a\geq 0 \Leftrightarrow \pi(a) \geq 0 \in B(\mathcal{H})[/itex]?